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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Dixmier-Moeglin equivalence
in quantum coordinate rings
and quantized Weyl algebras


Authors: K. R. Goodearl and E. S. Letzter
Journal: Trans. Amer. Math. Soc. 352 (2000), 1381-1403
MSC (1991): Primary 16S36, 16P40, 81R50
Published electronically: October 15, 1999
MathSciNet review: 1615971
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Abstract: We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of $n\times n$ matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras $A$ over infinite fields $k$ of arbitrary characteristic. Our main result is the verification, for $A$, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal $P$ of $A$ is primitive if and only if the center of the Goldie quotient ring of $A/P$ is algebraic over $k$, if and only if $P$ is a locally closed point - with respect to the Jacobson topology - in the prime spectrum of $A$. These equivalences are established with the aid of a suitable group $\mathcal{H} $ acting as automorphisms of $A$. The prime spectrum of $A$ is then partitioned into finitely many ``$\mathcal{H} $-strata'' (two prime ideals lie in the same $\mathcal{H} $-stratum if the intersections of their $\mathcal{H} $-orbits coincide), and we show that a prime ideal $P$ of $A$ is primitive exactly when $P$ is maximal within its $\mathcal{H} $-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which $\mathcal{H} $ acts transitively on the set of rational ideals within each $\mathcal{H} $-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of $A$. For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.


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Additional Information

K. R. Goodearl
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: goodearl@math.ucsb.edu

E. S. Letzter
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: letzter@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02345-4
PII: S 0002-9947(99)02345-4
Received by editor(s): August 16, 1997
Published electronically: October 15, 1999
Additional Notes: The research of the first author was partially supported by NSF grant DMS-9622876, and the research of the second author was partially supported by NSF grant DMS-9623579.
Article copyright: © Copyright 1999 American Mathematical Society